Riemann integral problem

teodandon

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Feb 4, 2016
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Hi, this is one of the problems my teacher gave us as preparation for the upcoming exam:
Define the notion of Riemann integral on [a1;b1]×[a2;b2].
I know the definition of both the Riemann integral and the sum, but I'm not sure how to apply it for this cartesian product.
 
Hi, this is one of the problems my teacher gave us as preparation for the upcoming exam:
Define the notion of Riemann integral on [a1;b1]×[a2;b2]. I know the definition of both the Riemann integral and the sum, but I'm not sure how to apply it for this Cartesian product.
Please note the capital, Cartesian comes a man's name. Welcome the the forum. It rule that posters show some work so that we have some idea of how to help. However I think this case may warrant an exception.
You say you know simple Riemann integrals. Then you realize that a closed interval is sub-divided into smaller intervals, arbitrary points are selected in each cell and the area of a rectangle is calculated. The sum of those areas is an approximating sum of the integral. So the same idea is applied in \(\displaystyle \Re^2\). There are many different approaches to how this is done, I don't know which your textbook/instructor will require. But all come down to filling the area with small rectangles and adding up their areas.
 
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